This paper is a corrected version of the paper of the same title given at the Western Joint Computer Conference, May 1961. A tenth section discussing the relations between mathematical logic and computation has been added. Programs that learn to modify their own behaviors require a way of representing algorithms so that interesting properties and interesting transformations of algorithms are simply represented. Theories of computability have been based on Turing machines, recursive factions of integers and computer programs. Each of these has artificialities which make it difficult to manipulate algorithms or to prove things about them. The present paper presents a formalism based on conditional forms and recursive functions whereby the functions computable in terms of certain base functions can be simply expressed. We also describe some of the formal properties of conditional forms and a method called recursion induction for proving facts about algorithms. A final section in the relations between computation and mathematical logic is included.